A general linear optimization algorithm based upon labeling and factorizing of basic paths on RPM network PublicDeposited

Descriptions

A new method is developed for solving linear optimization
problems based on the RPM network modeling technique which represents
the primal and the corresponding dual models simultaneously
upon a single graph. The network structure is used to eliminate
the need for explicit logical variables and to provide a graphic
tool in analyzing the problem.
The new algorithm iterates through a finite number of basic
solutions working towards optimality (primal) or towards feasibility
(dual). At each iteration a set of critical constraints and basic
structural variables are identified to form the current basic path
network. A solution for the basic variables is obtained through
factorization of the basis and used to update the nonbasic network.
If the Kuhn-Tucker conditions are not satisfied, the method proceeds
with the next iteration unless an unbounded or infeasible solution
is encountered.
Under the new scheme, the original data remains unchanged
throughout the optimization procedure and round-off errors can be
kept to a minimum. Furthermore, the basic paths representation used
in factorization reduces computer core requirement and permits
direct - addressing of pertinent non-basic node data on disk storage.
These features are especially appealing in solving large-scale
problems even on limited computer hardware.
Since the size of the basis is never greater than the size of
the basis required by simplex-type algorithms, the new scheme has an
advantageous memory storage requirement.
Any basic solution (not necessarily optimum or feasible) can be
used as a starting point and multipivoting can accelerate the
optimization process.
In general, the number of iterations and the amount of operations
depends on the sparsity of the constrained matrix and the complexity
of the problem.
Statistical data based on sample experimental results indicate
that the new algorithm, on the average, requires less arithmetic
operations and no more iterations to reach the final solution than
the simplex-type algorithms.